A new quasi-Coulomb relation between magnetic force vs distance, magnetic flux density, and magnetic pole area is proposed in this paper based on the empirical analysis on the magnetic force vs distance measurements. A simple device is designed and fabricated to measure the repulsive/attractive magnetic force between two magnets. The measurement results show that the relationship between the magnetic force and distance is similar to Coulomb’s electrostatic law, but the attraction is greater than the repulsion. The magnetic force is proportional to the integral of the magnetic flux density with the areas of the magnetic pole. This quasi-Coulomb relationship is useful to evaluate the torque and speed of a rim-driven motor, which plays an important role in electrified propulsion in green aviation technology.

## I. THE BACKGROUND

A rim-driven permanent magnet motor (RDM)^{1} is an advanced and more efficient electric propulsion technology, which has outstanding technical advantages in the field of electric aircraft propulsion.^{2,3} Instead of a shaft-driven propeller, the turbofan blade is attached inside the inner rim driven by the interactive magnetic force of the permanent magnet inside the rim. The electromagnetic force does not need to be transmitted to the blade through the central shaft, and the driving efficiency is higher. In accordance with Archimedes’ lever principle, rim-driven is more advantageous than shaft-driven as illustrated in Fig. 1. The inner fan blade is driven by the ring rather than the hub axis that drives the outstretching propeller. More powerful driving ability can be achieved to conquer the big torque induced by the large thrust force in aero-engine propulsion.

As shown in Fig. 2, the RDM consists of two rings connected by bearings. The outer ring is a stator made of an electromagnet controlled by commutating AC electric power, the inner ring is a rotor made of a permanent magnet, and the fan blade is attached with the inner rotor to generate thrust. The motor operates as follows:

The permanent magnet of the inner ring moves counterclockwise under the repulsion of electromagnet 1 and the attraction of electromagnet 2.

When the permanent magnet is about to pass over electromagnet 2, the Hall sensor detects the incoming magnet and control electromagnet 2 reverses the South Pole and North Pole, so that the permanent magnet continues to receive a repulsive force from electromagnet 2 and continues to move forward, in the meantime receiving the attraction from electromagnet 3. In this way, the permanent magnet together with the inner ring will continuously keep moving forward under the dual repulsive and attractive force from outer ring electromagnets.

Under the magnetic force of M1 and M2, the speed of magnet M0 is

where *F*_{M1} and *F*_{M2} are the combined magnetic forces applied on M0 (the tangent portion of the force, discussed in the following), and *F*_{R} is the dragging force, which can be equivalent to the thrust. $\u222bOPFdx$ represents the work done from the starting point O to point P.

The rotating speed and torque of the rim-driven motor are of great significance to evaluate the thrust of the aero-engine. The basic law of physics on the magnetic force between two magnets with distance, the area of the pole, and the magnetic intensity is necessary for this evaluation. It should be mentioned that such a basic scientific study is in fact still lacking even in traditional physics textbooks.^{4} This is the motivation of our work in this paper—an empirical model to quantify the magnetic force between two magnets vs distance, area, and magnetic density.

## II. THE RESULTS

### A. The empirical formula of magnetic force between two magnets

The magnetic force vs the spacing is measured using two neodymium magnet plates with different dimensions (*ϕ* 12 and *ϕ* 35 mm, thickness 5 mm). The attractive and repulsive magnetic forces follow the empirical formula:

All parameters follow SI units and B in Tesla. The empirical coefficients K and n are given in Table I.

. | . | ϕ 12 mm-diameter solid magnet disk
. | ϕ 35 mm-diameter ring magnet disk
. |
---|---|---|---|

K | Attractive | 5.83 × 10^{−9} | 1.34 × 10^{−8} |

Repulsive | 3.19 × 10^{−9} | 8.47 × 10^{−9} | |

n | Attractive | 1.04 | 0.85 |

Repulsive | 0.89 | 1 |

. | . | ϕ 12 mm-diameter solid magnet disk
. | ϕ 35 mm-diameter ring magnet disk
. |
---|---|---|---|

K | Attractive | 5.83 × 10^{−9} | 1.34 × 10^{−8} |

Repulsive | 3.19 × 10^{−9} | 8.47 × 10^{−9} | |

n | Attractive | 1.04 | 0.85 |

Repulsive | 0.89 | 1 |

As can be seen, the magnetic force between two magnetic disks is approximately inversely proportional to the distance with the power ratio n close to 1, and the overall K is 10^{−9}–10^{−8}. This empirical formula is the starting point, and more delicate work is still needed. The quasi-Coulomb relationship between magnetic force, flux density, pole area, and distance can be used to calculate the velocity of RDP (Rim-Driven Propulsion).

### B. The attractive force is larger than repulsive force

Unlike the Coulomb force applied on electrons, the attractive magnetic force is stronger than the repulsive force as shown in Fig. 3. In general, the coefficient K in Eq. (2) represents the difference between the attractive and repulsive forces, and the ratio is 1.5–1.8. For the *ϕ* 35 neodymium magnetic disk, the attractive magnetic force can be 5 N more than the repulsive one. Its magnetic force is too strong to control the distance during the measurement when the two disks are too close to each other. (It is almost impossible to separate two *ϕ* 35 neodymium magnetic disks once they are attracted together, special equipment is needed.)

As for the repulsive magnetic force, the North-to-North and South-to-South magnetic forces are the same.

Our tentative explanation that the attractive force is bigger than the repulsive force is as follows.

The magnetic dipole intends to align together when the North Pole is facing the South Pole. However, when the North Pole is facing the North Pole, or the South Pole is facing the South Pole, the magnetic dipole intends to repel each other in order to make the permanent magnet lose the permanent magnetism. The United States is always more powerful than the divided states fighting each other during the civil war.

Therefore, the attractive magnetic force is greater than the repulsive magnetic force because they belong to two different processes: forward enhancement and reverse torsion of the magnetism. As for electric charges, Coulomb’s force is equivalent for repulsive and attractive interactions between two single (not dipole) point charges.

## III. THE EXPERIMENT

### A. The magnetic flux distribution

The numerator in formula (2) is the product of magnetic field intensity and pole area. As the magnetic field distributes unevenly along the surface of the disk, the integration of the magnetic flux over the surface should be more accurate as this product. Therefore, at first, the distribution of magnetic flux on the surface is characterized with the nine-point measurement. We have measured 14 disks, and all the measured data are shown in Fig. 4. As can be seen, the magnetic flux values at the same distance from the center are similar, and the variation is within 3%.

Then, we carefully characterized the distribution of magnetic flux for different shapes of magnets—a solid magnet with a diameter of 12 mm and a ring-type neodymium permanent magnet with an outer diameter of 35 mm.

The magnetic flux distribution is shown in Fig. 5. The following can be seen.

For a solid magnet disk, the magnetic flux density is greater at the edge.

Although the center is hollow in the ring magnet, there is a high magnetic field strength at the center,

For the convex surface, the minimum magnetic flux is at the inner edge, reaches the maximum value at the outer edge, and then gradually falls down along the radial until the edge.

For the flat surface, the minimum magnetic flux is at the edge and then gradually increases along the radial until the edge.

According to the above distribution, the *BS* product is obtained with segmented integration of ∫*BdS* in Eq. (2).

### B. The apparatus to measure magnetic force vs distance

A special device is formulated as shown in Fig. 6(a) and fabricated in Fig. 6(b) in order to measure the magnetic force with various distances. One magnet is fixed on the wall, while the other magnet is connected to the tension gauge. The distance between the two magnets is controlled by the rotation of the long screw at the rear end. The spacing between the two magnets is measured using a plastic caliper.

## IV. DISCUSSION

Our work presented here is still primitive. There are few other points needed for the future studies:

At the current stage, as the first order of approximation, the interaction between two magnets in Eq. (1) can be treated as a two-point charge using formula (2) to evaluate the speed and touch of the RDP driving force. The magnetic force is equivalent to the cosine portion of the total magnetic forces in Fig. 2.

The tangential force. The above test provides an account of a theoretical procedure and practical apparatus used to investigate axial forces between opposing permanent magnets and how the mutual attraction and repulsive forces between the magnets vary with the air gap. Still, the empirical formula of the tangential force is more relevant to estimate the rotational motor action (Fig. 2). Therefore, it is worthwhile to study the tangential force of two magnets vs distance in our future studies. A special testing apparatus is formulated and test is under implementation to test the tangential force between two magnets.

The correlation of semi-Coulomb’s law between magnetic force and electric force. The empirical formula in Eq. (2) and Table I shows that the coefficient n is close to 1, but in Coulomb’s law, this n value is 2. However, in our test, this interaction is the surface charge to surface charge, not the point to point charge. It is worthwhile to correlate the magnetic force vs the electronic force regarding the coefficient n, i.e., to evaluate the magnetic force between two magnetic dipoles from the theoretical point of view. We are going to discuss these issues in a further report.

## V. CONCLUSION

In this paper, a new simple device is formulated and implemented to establish the functional relationship between the repulsive force/attraction of two magnets and distance, magnetic flux density, and magnetic pole area. According to the experimental results, a new quasi-Coulomb relationship theory between the magnetic force and distance is proposed. The measurement results show that the relationship between the magnetic force and distance is similar to Coulomb’s electrostatic law, but the attractive force is greater than the repulsive force. The newly developed magnetic force formula is of indispensable significance for further estimation of the performance of a rim-driven motor, an advanced promising propulsion engine for future electric aviation.

## AUTHOR DECLARATIONS

### Conflict of Interest

We have no conflicts of interest to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request. The data are not publicly available.